Introduction to Polar Coordinates

Key Concepts

Rectangular Coordinates vs. Polar Coordinates

• Rectangular Coordinates: Uses (x, y).
• Polar Coordinates: Uses (r, θ).
  • r: Radius of the circle.
  • θ (theta): Angle measured from the positive x-axis.

Plotting Polar Coordinates

Example 1: Plotting (3, 45°)

• r = 3, θ = 45°.
• Draw three circles with radii 1, 2, and 3.
• Draw a ray at a 45° angle extending to the third circle.
• Positive angles rotate counterclockwise from the x-axis.

Example 2: Plotting (2, 3π/4)

• Conversion: 3π/4 × (180/π) = 135°.
• Angle (θ) between 90° (Quadrant I) and 180° (Quadrant II).
• Plot to the 2nd circle at 135°.

Example 3: Plotting Negative r

• Example: (-2, 60°)
  • First plot (2, 60°): 2nd circle at 60°.
  • Negative r: Go 180° opposite; (-2, 60°) is same as (2, 240°).

Example 4: Plotting Another Negative r

• Example: (-3, 120°)
  • First plot (3, 120°): 3rd circle in Quadrant II.
  • Negative r: Go 180° opposite; (-3, 120°) is same as (3, -60°).

Generating Equivalent Coordinates

• Original Point: Example: (2, 30°).
• Other three equivalents within θ restriction (-360° to 360°):
  • Positive r, Negative θ: (2, -330°).
  • Negative r, Positive θ: (-2, 210°).
  • Negative r, Negative θ: (-2, -150°).

Formula Recap

• r: Always add/subtract 180° to/from θ.
• θ: Add/Subtract 360° based on positive/negative θ.

Converting Between Coordinate Systems

Polar to Rectangular

• Formulae:
  • x = r cos θ
  • y = r sin θ
• Example: (4, 60°)
  • x = 4 cos 60° = 2
  • y = 4 sin 60° = 2√3
• Example: (6, 5π/6)
  • x = 6 cos 5π/6 = -3√3
  • y = 6 sin 5π/6 = 3

Rectangular to Polar

• Formulae:
  • r = sqrt(x² + y²)
  • θ = arctan(y/x)
• Example: (2, -4)
  • r = √(2² + (-4)²) = 2√5
  • θ = atan(|-4/2|) = 63.4° (reference angle)
  • Quadrant IV: θ = 360° - 63.4° = 296.6°
• Example: (-5, 5√3)
  • r = √((-5)² + (5√3)²) = 10
  • θ = atan(√3) = 60°
  • Quadrant II: θ = 180° - 60° = 120°

Radian Equivalents

• 120° = 2π/3
• When converting: ensure r stays consistent with θ in radians.

Tips and Strategies

• Learn to generate additional points by systematically applying 180° or 360° transformations.
• Utilize calculator functions for accuracy in angle measurements.
• Practice with multiple examples to gain comfort with conversions and plotting.